If $S_1$ and $S_2$ are the foci of the hyperbola whose transverse axis length is 4 and conjugate axis length is 6, $S_1$ and $S_4$ are the foci of the conjugate hyperbola, then the area of the quadrilateral $S_1S_2S_3S_4$ is:
Step-by-Step Solution
Key Concept: For a hyperbola with semi-major axis a and semi-minor axis b, the eccentricity satisfies b² = a²(e² - 1). The conjugate hyperbola has swapped roles of a and b, creating a rectangle formed by four foci S₁, S₂, S₃, S₄ whose area equals 4 × (1/2) × ae × be₁, where e and e₁ are eccentricities of original and conjugate hyperbolas respectively.
The required area equals $4 \times \frac{1}{2}ae \cdot be_1 = 4 \times \frac{1}{2} \times 2 \times 3 \times ee_1$. From $b^2 = a^2(e^2 - 1)$, we get $e^2 = \frac{13}{4}$ and $e_1^2 = \frac{13}{9}$. The area becomes $12 \times \frac{\sqrt{13}}{2} \times \frac{\sqrt{13}}{3} = 2 \times 13 = 26$.
Correct Answer: 2